Abstract
In his work on F. Carlson's uniqueness theorem for entire functions of exponential type, Q. I. Rahman [5] was led to consider an infinite integral and needed to determine the rate at which the integrand had to go to zero for the integral to converge. He had an estimate for it which he was content with, although it was not the best that could be done. In the present paper we find a result about the behaviour of the integrand at infinity, which is essentially best possible. Stirling's formula for the Euler's Gamma function plays an important role in its proof.
Highlights
Let {λn}∞ −∞ be a sequence of real numbers such that |n−λn| ≤ L for all n ∈ Z and |λn − λm| ≥ 2δ > 0 for m = n
Let f (z) be an entire function and denote the maximum of |f (z)| on |z| = r by for some γ and that f (λn) = 0 for all n ∈ Z
Summary
Carlson’s theorem says (see [1, Chapter 9]) that if f is an entire function such that |f (z)| = O(eb|z|) as |z| → ∞ for some b < π and f (n) = 0 for n = 0, ±1, ±2, . Hadamard’s three circles theorem, Euler’s Gamma function. Qazi entire function and denote the maximum of |f (z)| on |z| = r by M (r). The proof of Theorem A is, in part, based on the following auxiliary result presented in [5] as Lemma 6.
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